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Pertanyaan

Jika x → 2 1 ​ lim ​ ⎝ ⎛ ​ x − 2 1 ​ 3 a x 3 + b ​ − 2 ​ ⎠ ⎞ ​ = A , maka nilai x → 2 1 ​ lim ​ ⎝ ⎛ ​ 4 x 2 − 1 3 8 a x 3 ​ + 8 b ​ ​ − 2 x ​ ⎠ ⎞ ​ = ....

Jika , maka nilai  

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A. Allamah

Master Teacher

Mahasiswa/Alumni Universitas Brawijaya

Jawaban terverifikasi

Jawaban

ka , maka nilai

 ka limit as x rightwards arrow 1 half of open parentheses fraction numerator cube root of a x cubed plus b end root minus 2 over denominator x minus begin display style 1 half end style end fraction close parentheses equals A, maka nilai limit as x rightwards arrow 1 half of open parentheses fraction numerator cube root of begin display style fraction numerator a x cubed over denominator 8 end fraction plus b over 8 end style end root minus 2 x over denominator 4 x squared minus begin display style 1 end style end fraction close parentheses equals fraction numerator A minus 4 over denominator 8 end fraction

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Pembahasan

Ingat Definisi turunan, jika Penyelesaian dapat menggunakan limit fungsi dengan metode L'Hospital dan menggunakan turunan Jika limit fungsi berbentuk , maka limit fungsi tersebut bisa diselesaikan dengan menggunakan turunan, yaitu Menggunakan sifat limit berikut Serta memisalkan , didapatkan Dengan demikian, dengan dalil L'Hospital diperoleh Akan dicari nilai dari . Dengan menjabarkan bentuk aljabar sebagai berikut Akan dicari nilai dengan L'Hospital. Jadi,nilai dari . Dengan demikian,ka , maka nilai

Ingat Definisi turunan, jika

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell a x to the power of n rightwards arrow f apostrophe open parentheses x close parentheses equals a n x to the power of n minus 1 end exponent end cell end table

Penyelesaian dapat menggunakan limit fungsi dengan metode L'Hospital dan menggunakan turunan

Jika limit fungsi berbentuk limit as x rightwards arrow k of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals 0 over 0, maka limit fungsi tersebut bisa diselesaikan dengan menggunakan turunan, yaitu

limit as x rightwards arrow k of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals limit as x rightwards arrow k of fraction numerator f apostrophe open parentheses x close parentheses over denominator g apostrophe open parentheses x close parentheses end fraction

Menggunakan sifat limit berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow k of f open parentheses x close parentheses plus-or-minus g open parentheses x close parentheses end cell equals cell limit as x rightwards arrow k of f open parentheses x close parentheses plus-or-minus limit as x rightwards arrow k of g open parentheses x close parentheses end cell row cell limit as x rightwards arrow k of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction end cell equals cell fraction numerator limit as x rightwards arrow k of f open parentheses x close parentheses over denominator limit as x rightwards arrow k of g open parentheses x close parentheses end fraction end cell row cell limit as x rightwards arrow k of a f open parentheses x close parentheses end cell equals cell a limit as x rightwards arrow k of f open parentheses x close parentheses end cell end table

Serta memisalkan f open parentheses x close parentheses equals cube root of a x cubed plus b end root, didapatkan

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 half of open parentheses fraction numerator f open parentheses x close parentheses minus 2 over denominator x minus begin display style 1 half end style end fraction close parentheses end cell equals cell limit as x rightwards arrow 1 half of open parentheses fraction numerator f apostrophe open parentheses x close parentheses over denominator 1 end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 1 half of f apostrophe open parentheses x close parentheses end cell row blank equals cell limit as x rightwards arrow 1 half of f apostrophe open parentheses x close parentheses end cell end table

Dengan demikian, dengan dalil L'Hospital diperoleh

 

table attributes columnalign right center left columnspacing 0px end attributes row blank blank blank row cell limit as x rightwards arrow 1 half of f apostrophe open parentheses x close parentheses end cell equals A end table

Akan dicari nilai dari limit as x rightwards arrow 1 half of open parentheses fraction numerator cube root of begin display style fraction numerator a x cubed over denominator 8 end fraction plus b over 8 end style end root minus 2 x over denominator 4 x squared minus begin display style 1 end style end fraction close parentheses. Dengan menjabarkan bentuk aljabar sebagai berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator cube root of begin display style fraction numerator a x cubed over denominator 8 end fraction plus b over 8 end style end root minus 2 x over denominator 4 x squared minus begin display style 1 end style end fraction end cell equals cell fraction numerator cube root of begin display style 1 over 8 open parentheses a x cubed plus b close parentheses end style end root minus 2 x over denominator 4 x squared minus begin display style 1 end style end fraction end cell row blank equals cell fraction numerator cube root of begin display style 1 over 8 end style end root times cube root of open parentheses a x cubed plus b close parentheses end root minus 2 x over denominator 4 x squared minus 1 end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style cube root of open parentheses a x cubed plus b close parentheses end root minus 2 x over denominator 4 x squared minus 1 end fraction end cell end table

Akan dicari nilai limit as x rightwards arrow 1 half of open parentheses fraction numerator cube root of begin display style fraction numerator a x cubed over denominator 8 end fraction plus b over 8 end style end root minus 2 x over denominator 4 x squared minus begin display style 1 end style end fraction close parentheses dengan L'Hospital.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 half of open parentheses fraction numerator cube root of begin display style fraction numerator a x cubed over denominator 8 end fraction plus b over 8 end style end root minus 2 x over denominator 4 x squared minus begin display style 1 end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 1 half of fraction numerator begin display style 1 half cube root of open parentheses a x cubed plus b close parentheses end root minus 2 x end style over denominator 4 x squared minus 1 end fraction end cell row blank equals cell limit as x rightwards arrow 1 half of fraction numerator begin display style 1 half f open parentheses x close parentheses minus 2 x end style over denominator 4 x squared minus 1 end fraction space end cell row blank equals cell limit as x rightwards arrow 1 half of fraction numerator begin display style 1 half f apostrophe open parentheses x close parentheses minus 2 end style over denominator 8 x end fraction space end cell row blank equals cell fraction numerator begin display style limit as x rightwards arrow 1 half of 1 half f apostrophe open parentheses x close parentheses minus limit as x rightwards arrow 1 half of 2 end style over denominator limit as x rightwards arrow 1 half of 8 x end fraction end cell row blank equals cell fraction numerator begin display style 1 half limit as x rightwards arrow 1 half of f apostrophe open parentheses x close parentheses minus 2 end style over denominator 8 limit as x rightwards arrow 1 half of x end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style A minus begin display style 2 end style over denominator 8 times begin display style 1 half end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half open parentheses A minus 4 close parentheses end style over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style A minus 4 end style over denominator 8 end fraction end cell end table

Jadi, nilai  darilimit as x rightwards arrow 1 half of open parentheses fraction numerator cube root of begin display style fraction numerator a x cubed over denominator 8 end fraction plus b over 8 end style end root minus 2 x over denominator 4 x squared minus begin display style 1 end style end fraction close parentheses equals fraction numerator A minus 4 over denominator 8 end fraction.

Dengan demikian, ka limit as x rightwards arrow 1 half of open parentheses fraction numerator cube root of a x cubed plus b end root minus 2 over denominator x minus begin display style 1 half end style end fraction close parentheses equals A, maka nilai limit as x rightwards arrow 1 half of open parentheses fraction numerator cube root of begin display style fraction numerator a x cubed over denominator 8 end fraction plus b over 8 end style end root minus 2 x over denominator 4 x squared minus begin display style 1 end style end fraction close parentheses equals fraction numerator A minus 4 over denominator 8 end fraction

Latihan Bab

Konsep Kilat

Konsep Limit

Sifat Limit

Limit Fungsi Aljabar

197

Dustin Nathanael

Ini yang aku cari!

Amanda Deby febrina

Makasih ❤️

A. Sitti Nurhalijah

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Dani

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